Math. Z. 180, 335-348 (t982) Mathematische Zeitschrift O Springer-Vertag 1982 Primes in Short Intervals Glyn Harman* Department of Mathematics, Royal Holloway College, Egham, Surrey TW20 OEX, United Kingdom 1. Introduction It was conjectured by Cram6r [1] that every interval of the form In, n +f(n) log2n] contains a prime for some f(n)~l as n~oe. Assuming the Riemann Hypothesis, Selberg [12] has shown that almost all intervals of the above form contain a prime providing f(n)--*oe with n. "Almost all" in this context indicates that the number of n<-_X for which the statement is false is o(X). Selberg's proof essentially gave a relationship between the density of zeros of ~(s) and the length of the interval. This was used by Montgomery (Chap. 14 of [11]) to show that, for almost all n, [n, n 1/5+~] contains a prime. The exponent 1/5 may be improved to 1/6 using the zero density estimate of ~< 7/12+e Huxley [6] which he obtained to show that p,+l-p, ~p, where Pn is the nth prime. This result on the difference between consecutive primes has been ~5/9+~ improved by Iwaniec and Jutila [8] to ~,, , and by Heath-Brown and Iwaniec [5] to p~1/2o+~. These last two results were obtained by a sieve method. We shall use similar arguments to prove the following result: Theorem. For almost all n, the interval In, n + n (all 0)+ ~) (1) contains a prime number. It is the hypothesis of Lemma 5 below which sets (1/10)+e as the limit of the present method. We shall in fact show that for almost all n the interval (1) contains >>n(l/~~ -~ primes. This exhibits a common feature of sieve results: we obtain a lower bound which is a fraction of the "expected" number of primes under consideration. 2. Outline of Method In Sect. 2-4 we use the following standard notation: P(z) = 1-I P, V(z) = 1-I (1 - l/p), p<z p<z * Written while the author held a London University Postgraduate Studentship 0025-5874/82/0180/0335/$02.80